What benefits can be got by applying Graph Convolutional Neural Network instead of ordinary CNN?
Generally speaking a graph CNN is applied to data represented by graphs, not images. 


a graph is a collection of nodes and edges connecting them.
an image is a 2D or 3D matrix, in which each element denotes a pixel in space


If your data are just images, or something similar (e.g. some fMRI data), you usually cannot benefit from graph CNN compared with usual CNN.

Sometimes, the class labels of your images may be organized in a graph-like  (or tree-like) structure. In that case, you may have a chance to benefit from graph CNN.